Local 3D Shape Descriptor
Transcript of Local 3D Shape Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Scale-Dependent/Invariant Local 3D Shape Descriptors
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Scale-space
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Feature detector
– The feature detector is primary part in computer vision
– A specific scale is determined by size of local window
To overcome the limitation
–
Construct an image scale-space – Convolve the image with Gaussian kernels of increasing standard deviation
Scale-space
– Allow us to detect not only the location of a local feature in an image
– But also intrinsic scale
Scale in 3D geometric data
– Construct a representation of the data at different scale
• Multi-resolution representation of a mesh: sensitive to the sampling of the original model• Use smoothing operator similar to 2D scale-space: lead to alternations in the global topology
of the geometric data
– Distance metric for modeling the geometry
• 3D Euclidean distance: lead to creation of erroneous features
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Geometric scale-space in range image
– Range image is already a dense and regular projection of a single view of the surfaceof the target 3D shape
Geometric scale-space
– Construct a normal map NNNN
• Triangulate the range image
• Compute a surface normal for each vertex
– Geodesic distance
• Encode accurately the scale-variability of the surface geometry
• Approximate the distance with the sum of Euclidean 3D distances
–
Build the geometric scale-space of a base normal map NNNN• By filtering the normal map with Gaussian kernels of increasing standard deviation
• : local window
Geometric Scale-Space
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: a list of vertex point in the range image
: is a 2D domain in RRRR2
2D normal map
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Detect 3D geometric corner
– For a point uuuu in the normal map NNNNσ at scale σ
– The corner response is computed using the Gram matrix
•
NNNNs : horizontal direction, NNNNt : vertical direction• : a free parameter that is empirically set for each particular feature detector
• : weighing factor of the points in the Gram matrix
– Detect corner points at each scale
• Spatial local maxima of the corner detector responses
– Prune corners lying along edge points
• Threshold : the second-order partial derivatives
– Search for local maxima of the corner detector responses across the scales
• The scale of the corners is intrinsic scale of each corner
Scale-Dependent Features
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Limitation of 3D shape descriptor
– Sampling density
– Size of descriptor’s support region
Dense and regular 2D descriptors
– Insensitive to the resolution of the input range images
Support size
– Can be determine by the scale information of corner
Local 3D Shape Descriptors
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Solution in this paper
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Determine local neighborhood
– Map the local neighborhood of a corner to a 2D domain
– Use exponential map
Exponential map
–
Map from the tangent space of a surface point to the surface itself (map 2D image to3D surface)
– The exponential map takes a vector wwww on the tangent plane and maps it to the point
on the geodesic curve at a distance of 1 from uuuu, or Exp(wwww)= Γ(1)
– Any point vvvv on the surface in the local neighborhood of uuuu can be mapped to uuuu’s
tangent plane (referred to as Log map) – Geodesic polar coordinates of vvvv
• Geodesic distance
• Polar angle of the tangent to the geodesic at uuuu
Local 3D Shape Descriptors
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Exponential map Geodesic polar angle
uuuu eeee1111
eeee2222
θ
vvvv
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Build a scale-dependent local 3D shape descriptor
– Use the geodesic polar coordinates
– : geodesic distance between uuuu and vvvv
– : polar angle of the tangent of the geodesic between uuuu and vvvv defined
relative to a fixed bases {eeee1,eeee2}
–
Radius of the descriptor • Set proportional to the inherent scale σ
Interpolate a geometric entity
– To construct a dense and regular representation of the neighborhood of uuuu at scale σ
– Geometric entity: surface normal
Scale-Dependent Local 3D Shape Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Rotation invariant shape descriptor
– Rotate the normal such that the normal at the center point uuuu points in the positive z
direction
– The resulting dense 2D descriptor is invariant up to a single rotation (in-plane rotation
on the tangent plane)
– Align the maximum principal curvature direction at uuuu to the horizontal axis eeee1
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Scale-dependent local 3D shape descriptor
Scale-invariant local 3D shape descriptor
–
Build a set of scale-dependent local 3D shape descriptors – Normalize each descriptor’s size to a constant radius
Assumption for scale-invariant shape descriptor
– The scales of local geometric structures relative to the global scale of a range image
remains constant as the global scale of a range image is altered
– This assumption holds as long as the geometry captured in the range image is rigid
and does not go under any deformation
Scale-Dependent/Invariant Local 3D Shape Descriptor
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for a scale-dependent corner at uuuu and with scale σ
for a scale-dependent corner at uuuu and with scale σ
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Scale-Dependent Local 3D Shape Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Pairwise registration
– Similarity Measure
– A and B are the set of points in the domain of and
Multiview registration
Image registration
– The process of transforming different sets of data into one coordinate system
– Data may be multiple photographs, data from different sensors, from different times,
or from different viewpoints.
– Registration is necessary in order to be able to compare or integrate the dataobtained from these different measurements.
Evaluation of Local 3D Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Pairwise registration
Evaluation of Local 3D Descriptor
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Range images with the same global scale using a
set of scale-dependent local 3D shape descriptor
Range images with inconsistent global scales using
a set of scale-invariant local 3D descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Multiview registration
Evaluation of Local 3D Descriptor
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Initial set of range images Approximate registration
obtained with our
framework
Registration refined with
multi-view ICP [25]
A water tight model using
a surface reconstruction
algorithm [26]
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Multiview registration
– Multiple objects• 15 views of the Buddha model
• 12 views of the armadillo
• 15 views of the dragon model
Evaluation of Local 3D Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Multiview registration
– Random global scale• 15 views of the Buddha and dragon models
• Random scale factor: 1 ~ 4
Evaluation of Local 3D Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
Multiview registration
– Random global scale• 15 views of the Buddha models
• 12 views of the armadillo models
• 15 views of the dragon models
• Random scale factor: 1 ~ 4
Evaluation of Local 3D Descriptor
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Scale-Dependent/Invariant Local 3D Shape Descriptors
[1] J. Novatnack and K. Nishino, “Scale-Dependent/Invariant Local 3D Shape
Descriptors for Fully Automatic Registration of Multiple Sets of Range Images,”In Proceedings of the 10th European Conference on Computer Vision
(ECCV2008) , pp. 440-453, 2008.
[2] J. Novatnack and K. Nishino, “Scale-Dependent 3D Geometric Features,” In
IEEE 11th International Conference on Computer Vision (ICCV2007) , 2007.
References
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